For Euclidean space $\mathbb{R}^n, \ n\geq1,$ the $k$-level dyadic partition $\Delta_k, \ k=0,1,...,$ is defined as the set of cubes in $\mathbb{R}^n$ of sidelength $2^{-k}$ and corners in the set $$\bigl( 2^{-k}(v_1,v_2,...,v_n):\ v_i\in\mathbb{Z},\ i=1,2,...,n \bigr),\ k=0,1,...,$$ https://en.wikipedia.org/wiki/Dyadic_cubes such that for each $k$ the following holds true $\cup_{A\in\Delta_k} A = \mathbb{R}^n.$
The important feature of the above dyadic partitions is that they possess a "nestedness" property, which means that each $A\subset\Delta_{k}$ can be written as a union of $2^n$ sets from $\Delta_{k+1}.$
My question is about dyadic partitions of more general sets possessing the above "nestedness" property. To be more precise, I am interested in partitions of $n$-dimensional spheres $\mathbb{S}^n.$ I found in [1, Theorem 2.1], that one can construct partitions of doubling metric spaces https://en.wikipedia.org/wiki/Doubling_space with the following properties:
If $X$ is a metric space with the finite doubling property and $0<r<\frac{1}{3}$, then there exists a collection {$Q_{k, i} : k \in \mathbb{Z}, i \in N_k \subset \mathbb{N}$} of Borel sets having the following properties:
- $X=\cup_{i\in N_k}Q_k,$ for every $k\in\mathbb{Z},$
- $Q_{k,i}\cap Q_{m,j}= \emptyset$ or $Q_{k,i}\subset Q_{m,i}$ where $k,m\in\mathbb{Z}, \ k\geq m, \ i\in N_k$ and $j\in N_m,$
- for every $k\in\mathbb{Z}$ and $i\in N_k$ there exists a point $x_{k,i}\in X$ so that $ B(x_{k,i}, cr^k) \subset Q_{k,i} \subset B(x_{k,i}, Cr^k),$ where $c=\frac{1}{2}-\frac{r}{1-r}$ and $C=\frac{1}{1-r}.$
As $\mathbb{S}^n$ endowed with Euclidean distance is a doubling metric space, one can adapt the above approach to construct partitions of $\mathbb{S}^n.$ However, that construction does not possess "nestedness" property in its full generality as it states that if $A\in\Delta_k,$ then it can be represented as a union of sets from $\Delta_{k+1},$ but the number of sets in the union is unknown. My question: is it possible to construct partitions $\Delta_k, \ k=0,1,...,$ of $\mathbb{S}^n$ such that the following "nestedness" property remains true and what those partitions may look like:
Each $A\subset\Delta_{k}$ can be written as a union of $C^n$ sets from $\Delta_{k+1},$ where $C\in\mathbb{N}$ is some integer.
References:
[1] Käenmäki, A., Tapio, R., and Ville S. "Existence of doubling measures via generalised nested cubes." Proceedings of the American Mathematical Society 140.9 (2012): 3275-3281.